49x^2-77x+30 eching

$$49 x ^ { 2 } - 77 x + 30$$

Omil

$$\left(7x-6\right)\left(7x-5\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 49x^{2}+ax+bx+30 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-77 ab=49\times 30=1470$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 1470-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-1470 -2,-735 -3,-490 -5,-294 -6,-245 -7,-210 -10,-147 -14,-105 -15,-98 -21,-70 -30,-49 -35,-42$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-1470=-1471 -2-735=-737 -3-490=-493 -5-294=-299 -6-245=-251 -7-210=-217 -10-147=-157 -14-105=-119 -15-98=-113 -21-70=-91 -30-49=-79 -35-42=-77$$

Yechim – -77 yigʻindisini beruvchi juftlik.

$$a=-42 b=-35$$

49x^{2}-77x+30 ni \left(49x^{2}-42x\right)+\left(-35x+30\right) sifatida qaytadan yozish.

$$\left(49x^{2}-42x\right)+\left(-35x+30\right)$$

Birinchi guruhda 7x ni va ikkinchi guruhda -5 ni faktordan chiqaring.

$$7x\left(7x-6\right)-5\left(7x-6\right)$$

Distributiv funktsiyasidan foydalangan holda 7x-6 umumiy terminini chiqaring.

$$\left(7x-6\right)\left(7x-5\right)$$

Kvadrat tenglamalaridan foydalanish qoidalari

Kvadrat koʻp tenglama bu orqali hisoblanadi: ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), bu yerda x_{1} va x_{2} ax^{2}+bx+c=0 kvadrat tenglamaning yechimlari.

$$49x^{2}-77x+30=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$x=\frac{-\left(-77\right)±\sqrt{\left(-77\right)^{2}-4\times 49\times 30}}{2\times 49}$$

-77 kvadratini chiqarish.

$$x=\frac{-\left(-77\right)±\sqrt{5929-4\times 49\times 30}}{2\times 49}$$

-4 ni 49 marotabaga ko'paytirish.

$$x=\frac{-\left(-77\right)±\sqrt{5929-196\times 30}}{2\times 49}$$

-196 ni 30 marotabaga ko'paytirish.

$$x=\frac{-\left(-77\right)±\sqrt{5929-5880}}{2\times 49}$$

5929 ni -5880 ga qo'shish.

$$x=\frac{-\left(-77\right)±\sqrt{49}}{2\times 49}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-77\right)±7}{2\times 49}$$

-77 ning teskarisi 77 ga teng.

$$x=\frac{77±7}{2\times 49}$$

2 ni 49 marotabaga ko'paytirish.

$$x=\frac{77±7}{98}$$

x=\frac{77±7}{98} tenglamasini yeching, bunda ± musbat. 77 ni 7 ga qo'shish.

$$x=\frac{84}{98}$$

\frac{84}{98} ulushini 14 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{6}{7}$$

x=\frac{77±7}{98} tenglamasini yeching, bunda ± manfiy. 77 dan 7 ni ayirish.

$$x=\frac{70}{98}$$

\frac{70}{98} ulushini 14 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{5}{7}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{6}{7} ga va x_{2} uchun \frac{5}{7} ga bo‘ling.

$$49x^{2}-77x+30=49\left(x-\frac{6}{7}\right)\left(x-\frac{5}{7}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{6}{7} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$49x^{2}-77x+30=49\times \frac{7x-6}{7}\left(x-\frac{5}{7}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{5}{7} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$49x^{2}-77x+30=49\times \frac{7x-6}{7}\times \frac{7x-5}{7}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{7x-6}{7} ni \frac{7x-5}{7} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$49x^{2}-77x+30=49\times \frac{\left(7x-6\right)\left(7x-5\right)}{7\times 7}$$

7 ni 7 marotabaga ko'paytirish.

$$49x^{2}-77x+30=49\times \frac{\left(7x-6\right)\left(7x-5\right)}{49}$$

49 va 49 ichida eng katta umumiy 49 faktorini bekor qiling.

$$49x^{2}-77x+30=\left(7x-6\right)\left(7x-5\right)$$

Baholash

$$\left(7x-6\right)\left(7x-5\right)$$